Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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10 Most Recent Blog Comments:

Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 18, 2021 at 12:03am

#1019575

Speling Iz Dificult

Ever wonder why spelling is so damn hard?

Typos, tricks and misprints
Why is English spelling so weird and unpredictable? Don’t blame the mix of languages; look to quirks of timing and technology

Yeah, I don't know about that, a bunch of stolen words from other languages has to take at least part of the blame.

Part of the problem is that English spelling looks deceptively similar to other languages that use the same alphabet but in a much more consistent way. You can spend an afternoon familiarising yourself with the pronunciation rules of Italian, Spanish, German, Swedish, Hungarian, Lithuanian, Polish and many others, and credibly read out a text in that language, even if you don’t understand it. Your pronunciation might be terrible, and the pace, stress and rhythm would be completely off, and no one would mistake you for a native speaker – but you could do it. Even French, notorious for the spelling challenges it presents learners, is consistent enough to meet the bar. There are lots of silent letters, but they’re in predictable places. French has plenty of rules, and exceptions to those rules, but they can all be listed on a reasonable number of pages.

Sure, but consider the French translation of "birds:" oiseaux. Not one single solitary goddamned letter in that word is pronounced the way it ought to be pronounced.

The answer to the weirdness of English has to do with the timing of technology. The rise of printing caught English at a moment when the norms linking spoken and written language were up for grabs, and so could be hijacked by diverse forces and imperatives that didn’t coordinate with each other, or cohere, or even have any distinct goals at all. If the printing press had arrived earlier in the life of English, or later, after some of the upheaval had settled, things might have ended up differently.

Okay. This author is a linguist so I'm inclined to give more weight to what they're saying than, for example, what I'm thinking.

I don't have much more to say about the article, but it's a fascinating brief history of written English.

I will say this, though: At some point, the spelling/pronunciation link becomes a shibboleth. I think people use it to identify in-groups. For example, in my area, there's a road with the name Rio Road. We use it to spot tourists. "Yeah, I hear (business) is on Ree-oh Road." Oh, they must be from out of town; the proper pronunciation is Rye-oh. Or there's a nearby town named Staunton. You hear someone pronounce it "stawn-ton" and you know they ain't from around here and need to be watched carefully and maybe lynched.

Probably the worst offender in the orthography world, though, is the geoduck.

You see a word like that, and you think: oh, it must be gee-oh-duck. And it's probably a bird, right? An... earth bird? Well, obviously you're an ignorant rube and unworthy of respect because you didn't know it's pronounced "gooeyduck" and it's actually an enormous mollusk. How in the inconsistent hell do you get "gooey" from "geo?" I mean, seriously, goddamn, STOP IT.

So my contention is a language's inconsistencies are mostly there to fool outsiders to the language. It's like a secret code between club members: there to distinguish the cool people from the out-of-touch. You can also identify nerds with it: people who read more than they ought to tend to pronounce words they way they sound in their heads. For example, I know someone who, upon seeing a picturesque scene, always called it "picture-skew." Because that's a perfectly legitimate reading of the pronunciation of all those letters.

Now, I'm not one of the long line of idiots who try (and certainly fail) to "reform" English spelling to something more consistent. Such efforts were always doomed to failure, because, as I noted, it's a private club and you can't get in until you familiarize yourself with, at least, 90% of the silly rules, and even then, we'll still mark you as an outsider because you'll certainly have an accent. But dammit, I'm tempted.

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What's your favorite example of a word that ought not to be spelled like it is? Comment here before midnight tonight WDC time (Monday night) and you might get a Merit Badge on Tuesday.

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